A False Discovery Rate Control Method Using a Fully Connected Hidden Markov Random Field for Neuroimaging Data

False discovery rate (FDR) control methods are essential for voxel-wise multiple testing in neuroimaging data analysis, where hundreds of thousands or even millions of tests are conducted to detect brain regions associated with disease-related changes.

The proposed fcHMRF-LIS method is a novel spatial FDR control method that integrates the Local Index of Significance (LIS)-based testing procedure with a fully connected Hidden Markov Random Field (fcHMRF) to address complex spatial dependencies, maintain low variability in false discovery proportion (FDP) and false non-discovery proportion (FNP), and achieve computational scalability.

Problem Formulation and LIS-based Testing

The method is designed to infer the unknown hypothesis states h from observed test statistics x using the Local Index of Significance (LIS). The LIS for a hypothesis Hi is defined as p(hi = 0x) (Equation 1), which is the conditional probability that the null hypothesis is true given all test statistics x. The LIS-based testing procedure rejects hypotheses based on the ranked LIS values, specifically rejecting all H(i) with i = 1 to k, where k is defined by max[j: 1/j X j i=1 LIS(i)(x) ≤ α] (Equation 2). This procedure is valid for controlling FDR at level α and is asymptotically optimal for minimizing the FNR under mild conditions.

The fcHMRF Model

The core of the method is the novel fully connected HMRF designed to model complex spatial structures. The model assumes that observed test statistics x are conditionally independent given the hidden states h = (h1,..., hm), leading to a conditional probability density function p(xh) defined by Equation 3: p(xh) = Ym i=1 p(xi hi) with p(xi hi) = (1 − hi)f0(xi) + hi f1(xi). The unknown non-null density f1 is estimated nonparametrically using kernel density estimation. The fcHMRF models the probability of hidden states h by a fully connected Markov random field, where pairwise potentials wi j modulate the penalty based on spatial distance and mean difference: wi j = w1 exp(− X s∈S li,s − l j,s 2θ2 α,s − ∆µi − ∆µj2 / 2θ2 β) + w2 exp(− X s∈S li,s − l j,s 2θ2 γ,s).

Efficient Parameter Estimation via EM Algorithm

Since exact inference in the fcHMRF is intractable, the method employs an Expectation-Maximization (EM) algorithm to estimate the model parameters ϕ = [f1, w]. To make this process computationally scalable, it incorporates three key techniques:

1. The mean-field approximation to compute LIS estimates by approximating p(hx; ϕ) with q(hx; ϕ).

2. The CRF-RNN technique to unroll the mean-field iterations into differentiable operations analogous to neural network layers, allowing parameter sets ψ to be learned via backpropagation.

3. Permutohedral lattice filtering to accelerate the message-passing step within each mean-field iteration, reducing time complexity from quadratic to linear in the number of tests.

Simulation and Application Results

Extensive simulations compared fcHMRF-LIS against eight existing FDR control methods, demonstrating its superiority in several aspects:

(a) FDR Control:

(b) Stability:

The results show that fcHMRF-LIS achieves accurate FDR control, lower FNR, higher TP, and reduced variability across replications compared to existing methods. It successfully controls the empirical FDR at the nominal level in all simulation settings.

ADNI Data Analysis

Applied to an FDG-PET dataset from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) with 439,758 voxels, fcHMRF-LIS identified neurobiologically relevant brain regions. The method achieved significantly improved computational efficiency, requiring only about 1.2 hours on a 20-core CPU server without GPU acceleration, which is significantly faster than methods like LAWS or nnHMRF-LIS when using GPUs. Furthermore, in the most challenging EMCI2AD vs. CN comparison, fcHMRF-LIS identified 20 ROIs supported by prior structural and functional MRI studies that were missed by classical FDR control methods using the FDG-PET data.

Conclusion

fcHMRF-LIS is proposed as a powerful, stable, and scalable spatial FDR control method for voxel-wise multiple testing in neuroimaging data.

Based on the provided scientific paper, here are specific improvements that could be made to AI systems, categorized by the capabilities they would gain:

) Improve spatial dependency modeling in deep learning models (like NeuralFDR and DeepFDR).

• By integrating the fully connected Hidden Markov Random Field (fcHMRF) structure into the loss function or as a prior in a neural network architecture, AI systems can move beyond simple local neighborhood assumptions.

• This allows the system to explicitly model long-range interactions and spatial heterogeneity (captured by the appearance and smoothness kernels) when performing multiple hypothesis testing on neuroimaging data.

• The improved AI system could perform Neurobiological Localization with higher confidence, as it would better understand how distant brain regions are statistically related, leading to more biologically plausible hypotheses about disease progression.

) Enhance stability and reliability in unsupervised high-dimensional inference tasks.

• By adopting the fcHMRF-LIS framework (which uses the LIS procedure guided by a parsimonious model), AI systems can achieve lower variance in their discovery proportions (FDP/FNP) across replications compared to methods like DeepFDR or LAWS.

• This leads to more robust and trustworthy results for exploratory analyses where ground-truth states are unknown. The AI system would provide more consistent outputs even when trained on limited data or facing noisy input, minimizing the risk of spurious discoveries.

) Achieve computational scalability for massive, high-resolution datasets without prohibitive time costs.

• The integration of the CRF-RNN technique (CRF as Recurrent Neural Network) and permutohedral lattice filtering reduces the time complexity from quadratic to linear in the number of tests.

• This allows AI systems to analyze extremely large neuroimaging datasets (like the 439,758 voxels in ADNI) efficiently, enabling real-time or near real-time analysis for clinical applications where computational resources are constrained.

• The improved system can process high-resolution data with significantly lower latency than existing methods (e.g., reducing runtime by a factor of 10 or more compared to neural network approaches), making large-scale longitudinal studies feasible on standard CPU hardware.

) Optimize parameter estimation in complex, unsupervised models using advanced optimization techniques.

• By developing an efficient Expectation-Maximization (EM) algorithm accelerated by mean-field approximations and gradient descent via backpropagation, AI systems can automatically learn the underlying spatial parameters of the fcHMRF model (i.e., the weights of the appearance and smoothness kernels).

• The improved AI system would be capable of self-configuring its spatial modeling based on data characteristics, rather than relying on manually tuned hyperparameters, leading to more adaptive and accurate inference models for novel neuroimaging datasets.

) Develop a unified framework for sequential or spatial data analysis.

• The paper's extension potential suggests using this framework for other complex structures like GWAS SNP data and time series. An AI system built on fcHMRF could be generalized to handle these dependencies, allowing it to infer latent states in sequential or spatially correlated biological markers (e.g., inferring disease progression trajectories from longitudinal imaging scans).

• This would enable the AI to model complex biological processes where the state at time/location 1 is probabilistically dependent on the state at time/location 2, providing a holistic view of disease dynamics.